A KnowledgeSensitive Online Exercise for Developing Algebraic
Calculation Strategies
Hitoshi Nishizawa
nisizawa@toyotact.ac.jp
Electrical & Electronics Engineering
Toyota National College of Technology
Japan
Takayoshi YOSHIOKA
yoshioka@toyotact.ac.jp
Electrical and Electronic Engineering
Toyota National College of Technology
Japan
Karl Fuchs
Salzburg University
Austria
Alfred Dominik
yoshioka@toyotact.ac.jp
BORG Akademiestrasse
Austria
Abstract
Our
observations indicated that students did not learn the calculation of
algebraic rational expressions easily even if the students had enough
skills to calculate numerical fractions. We suspected the reason is that
algebraic calculations depend on several mathematical rewriting rules,
which are usually hidden in black boxes at numerical calculations, and
the students can not use the rules properly. The students can calculate
numerical fractions automatically according to a few patterns they have
learnt from experience and without thinking of the mechanism of the calculating
steps. They can calculate even polynomials in such a way. They are surprised
to find that their patternmatching algorithms often fail in the calculation
of rational expressions. Their patterns sometimes lead them to typical
calculating errors lead by incorrect calculating rules, but they can not
separate such kind of serious errors from other errors caused by careless
manipulation.
Teaching rulebased calculation in a class was not enough for these students.
They thought that the rules were too obvious to learn and they knew the
rules already. But knowing the rules and using them properly are separate
things. Traditional exercises of calculations improved the students' manipulating
skills and mostly decreased the errors caused by carelessness, but they
did not improve the students' understanding on mathematical rewriting
rules very much. The students, who could not separate the errors caused
by incorrect rules from careless mistakes, were simply satisfied by the
fewer but still existing ratio of the errors. They were repeating the
errors lead by incorrect rewriting rules.
In this paper, the authors propose a new online exercise for such students
to learn the calculation of algebraic rational expressions. In the exercise,
a series of calculating steps from the question to the answer are displayed
on a student's screen. Some of the steps may include calculating errors,
and the others do not. The student is asked to examine every calculating
step, determine the type of the error for an erroneous step, and describe
the rewriting rule of a correct step.
Such an exercise, when it was done on a pencilandpaper basis, measured
the students' sensitivities to calculating errors and their affinities
to rulebased calculation.
The new exercise, if it is well organized with a manipulating exercise
and an interactive instruction, is expected to help the students, who
do not recognize the importance of mathematical rules in calculations,
to develop the strategies themselves. We expect the system to help the
students who used to improve very slowly in algebraic calculations even
after a series of traditional calculating exercises.
